Consider the planes `3x-6y+2z+5=0a n d4x-12y+3z=3.` The plane `67 x-162 y+47 z+44=0` bisects the angel between the given planes which a. contains origin b. is acute c. is obtuse d. none of these

Options

A. contains origin
B. is acute
C. is obtuse
D. none of these

Explanation

To solve the problem of determining the nature of the angle bisector plane given by the equation \(67x - 162y + 47z + 44 = 0\) between the two planes \(3x - 6y + 2z + 5 = 0\) and \(4x - 12y + 3z = 3\), we will follow these steps:

### Step 1: Write the equations of the planes
The equations of the two planes are:
1. Plane 1: \(3x - 6y + 2z + 5 = 0\)
2. Plane 2: \(4x - 12y + 3z - 3 = 0\) (rewritten as \(4x - 12y + 3z + (-3) = 0\))

### Step 2: Normalize the equations
To apply the angle bisector formula, we need to normalize the equations of the planes. The normal vector for Plane 1 is \((3, -6, 2)\) and for Plane 2 is \((4, -12, 3)\).

### Step 3: Calculate the magnitudes of the normal vectors
- For Plane 1:
\[ \| \mathbf{n_1} \| = \sqrt{3^2 + (-6)^2 + 2^2} = \sqrt{9 + 36 + 4} = \sqrt{49} = 7 \]
- For Plane 2:
\[ \| \mathbf{n_2} \| = \sqrt{4^2 + (-12)^2 + 3^2} = \sqrt{16 + 144 + 9} = \sqrt{169} = 13 \]

### Step 4: Apply the angle bisector formula
The angle bisector plane can be expressed as:
\[ \frac{3x - 6y + 2z + 5}{7} = \pm \frac{4x - 12y + 3z - 3}{13} \]

### Step 5: Cross-multiply and simplify
Cross-multiplying gives:
\[ 13(3x - 6y + 2z + 5) = \pm 7(4x - 12y + 3z - 3) \]

### Step 6: Expand both sides
Expanding both sides:
1. Left side:
\[ 39x - 78y + 26z + 65 \]
2. Right side (using the positive case):
\[ 28x - 84y + 21z - 21 \]

### Step 7: Rearranging the equation
Setting both sides equal gives:
\[ 39x - 78y + 26z + 65 = 28x - 84y + 21z - 21 \]
Rearranging leads to:
\[ (39 - 28)x + (-78 + 84)y + (26 - 21)z + (65 + 21) = 0 \]
This simplifies to:
\[ 11x + 6y + 5z + 86 = 0 \]

### Step 8: Compare with the given bisector plane
Now we compare this with the given bisector plane \(67x - 162y + 47z + 44 = 0\).

### Step 9: Determine if it contains the origin
To check if the bisector plane contains the origin, substitute \(x = 0\), \(y = 0\), \(z = 0\) into the equation:
\[ 67(0) - 162(0) + 47(0) + 44 = 44 \neq 0 \]
Thus, the plane does not contain the origin.

### Step 10: Determine if the angle is acute or obtuse
To determine if the angle is acute or obtuse, we check the sign of the product of the coefficients of the normal vectors:
\[ 3 \cdot 4 + (-6)(-12) + 2 \cdot 3 = 12 + 72 + 6 = 90 \]
Since \(90 > 0\), the angle between the planes is acute.

### Conclusion
The bisector plane does not contain the origin and the angle is acute. Therefore, the correct answers are:
- a. contains origin: **No**
- b. is acute: **Yes**
- c. is obtuse: **No**
- d. none of these: **No**